306 XV NORMED ALGEBRAS AND SPECTRAL THEORY If also A has a unit element e and is not the zero algebra (in other words, if e 7^ 0) we shall always assume that the norm satisfies the additional condi- tion (15.1.2) \\e\\ = 1. The inequality (15.1.1) shows that the bilinear mapping (x, y)\-+xy of A x A into A is continuous (5.5.1). A complete normed algebra (i.e., a normed algebra in which the under- lying normed vector space is a Banach space) is called a Banach algebra. It is clear that every subalgebra B of a normed algebra A (if A has a unit element e, we require that B contains e), endowed with the restriction to B of the norm ||jc||, is a normed algebra. If m is a closed two-sided ideal of A, then the quotient algebra A/m, endowed with the norm (12.14.10.1) induced by ||jc||, is again a normed algebra. For if x, y are two elements of A/m, then for each e > 0 there exist x e x and y e y such that from which it follows that ||jcy|| ^ (||jc|| + fi)(||j>|| + e); since xy e xy and since e was arbitrary, this shows that \\xy\\ rg \\x\\ - \\y\\. If moreover A has a unit element e, and if m + A, then e is the unit element of A/m and is ^0, hence \\e\\ g \\e\\ = 1 by definition, and on the other hand \\e\\ = \\e2\\ ^ \\e\\2, which implies that \\e\\ ^ 1. Hence \\e\\ = 1, and so A/m is a normed algebra. (15.1.3) Let A be a normed algebra. Then the closure in A of a subalgebra of A (resp. of a commutative subalgebra, a left ideal, a right ideal) is a sub- algebra (resp. a commutative subalgebra, a left ideal, a right ideal). By virtue of (5.4.1) and the principle of extension of identities, this follows directly from the continuity of multiplication in A, by the same proof as in (5.4.1). If A, B are two normed algebras, an algebra isomorphism u : A -+ B is said to be a topological isomorphism if it is bicontinuous, that is if (5.5.1 ) there exist two real numbers a > 0 and b > 0 such that a \\x\\ <| ||H(JC)|| ^ b \\x\\ for all x e A (5.5.1 ). The isomorphism u is said to be isometric if in addition we have ||M(JC)|| = \\x\\ for all xe A.pectral theory;